1.
TAGUCHI DESIGN OF
EXPERIMENTS
Prof. Charlton S. Inao
2.
• 2.2.1 Definition
• Taguchi has envisaged a new method of conducting
the design of experiments which are based on well
defined guidelines. This method uses a special set of
arrays called orthogonal arrays. These standard arrays
stipulates the way of conducting the minimal number
of experiments which could give the full information of
all the factors that affect the performance parameter.
The crux of the orthogonal arrays method lies in
choosing the level combinations of the input design
variables for each experiment.
3.
Assumptions of the Taguchi method
• The additive assumption implies that the individual or
main effects of the independent variables on
performance parameter are separable. Under this
assumption, the effect of each factor can be linear,
quadratic or of higher order, but the model assumes
that there exists no cross product effects (interactions)
among the individual factors. That means the effect of
independent variable 1 on performance parameter
does not depend on the different level settings of any
other independent variables and vice versa. If at
anytime, this assumption is violated, then the
additivity of the main effects does not hold, and the
variables interact.
4.
Designing an experiment
• The design of an experiment involves the
following steps
1. Selection of independent variables
2. Selection of number of level settings for each
independent variable
3. Selection of orthogonal array
4. Assigning the independent variables to each
column
5. Conducting the experiments
6. Analyzing the data Inference
5.
Selection of the independent
variables
• Before conducting the experiment, the
knowledge of the product/process under
investigation is of prime importance for
identifying the factors likely to influence the
outcome. In order to compile a
comprehensive list of factors, the input to the
experiment is generally obtained from all the
people involved in the project.
6.
Deciding the number of levels
• Once the independent variables are decided, the number of levels
for each variable is decided. The selection of number of levels
depends on how the performance parameter is affected due to
different level settings. If the performance parameter is a linear
function of the independent variable, then the number of level
setting shall be 2. However, if the independent variable is not
linearly related, then one could go for 3, 4 or higher levels
depending on whether the relationship is quadratic, cubic or higher
order.
In the absence of exact nature of relationship between the
independent variable and the performance parameter, one could
choose 2 level settings. After analyzing the experimental data, one
can decide whether the assumption of level setting is right or not
based on the percent contribution and the error calculations.
7.
Selection of an orthogonal array
•
•
Before selecting the orthogonal array, the minimum number of
experiments to be conducted shall be fixed based on the total number of
degrees of freedom [5] present in the study. The minimum number of
experiments that must be run to study the factors shall be more than the
total degrees of freedom available. In counting the total degrees of
freedom the investigator commits 1 degree of freedom to the overall
mean of the response under study. The number of degrees of freedom
associated with each factor under study equals one less than the number
of levels available for that factor. Hence the total degrees of freedom
without interaction effect is 1 + as already given by equation 2.1. For
example, in case of 11 independent variables, each having 2 levels, the
total degrees of freedom is 12. Hence the selected orthogonal array shall
have at least 12 experiments. An L12 orthogonal satisfies this
requirement.
Once the minimum number of experiments is decided, the further
selection of orthogonal array is based on the number of independent
variables and number of factor levels for each independent variable.
8.
Assigning the independent variables
to columns
• The order in which the independent variables are
assigned to the vertical column is very essential. In
case of mixed level variables and interaction between
variables, the variables are to be assigned at right
columns as stipulated by the orthogonal array [3].
• Finally, before conducting the experiment, the actual
level values of each design variable shall be decided. It
shall be noted that the significance and the percent
contribution of the independent variables changes
depending on the level values assigned. It is the
designers responsibility to set proper level values.
9.
Conducting the experiment
• Once the orthogonal array is selected, the
experiments are conducted as per the level
combinations. It is necessary that all the
experiments be conducted. The interaction
columns and dummy variable columns shall not
be considered for conducting the experiment,
but are needed while analyzing the data to
understand the interaction effect. The
performance parameter under study is noted
down for each experiment to conduct the
sensitivity analysis.
10.
Analysis of the data
•
•
Since each experiment is the combination of different factor levels, it is essential
to segregate the individual effect of independent variables. This can be done by
summing up the performance parameter values for the corresponding level
settings. For example, in order to find out the main effect of level 1 setting of the
independent variable 2 (refer Table 2.1), sum the performance parameter values
of the experiments 1, 4 and 7. Similarly for level 2, sum the experimental results of
2, 5 and 7 and so on.
Once the mean value of each level of a particular independent variable is
calculated, the sum of square of deviation of each of the mean value from the
grand mean value is calculated. This sum of square deviation of a particular
variable indicates whether the performance parameter is sensitive to the change
in level setting. If the sum of square deviation is close to zero or insignificant, one
may conclude that the design variables is not influencing the performance of the
process. In other words, by conducting the sensitivity analysis, and performing
analysis of variance (ANOVA), one can decide which independent factor dominates
over other and the percentage contribution of that particular independent
variable. The details of analysis of variance is dealt in chapter 5.
11.
Inference
• From the above experimental analysis, it is clear that the
higher the value of sum of square of an independent
variable, the more it has influence on the performance
parameter. One can also calculate the ratio of individual
sum of square of a particular independent variable to the
total sum of squares of all the variables. This ratio gives the
percent contribution of the independent variable on the
performance parameter.
• In addition to above, one could find the near optimal
solution to the problem. This near optimum value may not
be the global optimal solution. However, the solution can
be used as an initial / starting value for the standard
optimization technique.
12.
• Once the experiments are conducted, the program automatically
stores the process parameters and the corresponding experiment
number and level combination of all the design variables in the
blackboard. This raw data has been processed further to segregate
the main effect of each individual variable. The following are the
important parameters which the program automatically calculates.
• i) Mean value of each level of a design variable
• ii) Sum of square value of the design variables
• iii) Total sum of square
• iv) Percent contribution
• v) Near optimal value of the objective function
• vi) Confirmation test
• vii) ANOVA (Analysis of Variance) test
13.
• It shall be noted that the grand mean of all the
experiments is the same as the average of the
mean values of each level of a design variable as
shown in Figure 5.5. Based on the mean values of
each design variable, the sensitivity analysis is
performed.
• Sum of square value
• The sum of square of individual design variable
can be calculated using either of the following
equations
14.
• where L is the number of level, N is the number
of experiments conducted, R is the no of
repetition per level which equals , T is the sum of
process parameters of all the experiments, ......is
the grand mean value of all the experiments
which equals , and ...... is the mean value of jth
level value of ith variable.
• In case of L9 array which is given in Table 2.1, the
total sum of square of variable 3 can be
calculated using the equation 5.5 or 5.6.
15.
• Similarly the sum of square values for other
variables can also be found.
Total sum of square
The total sum of square (SSTO) is the sum of deviation of the experimental process
parameters from the grand mean value of the experiment. This can be obtained from
the equations 5.7 and 5.8.
16.
where .... is the performance parameters for
the kth experiment.
17.
•
•
•
•
•
•
•
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This total sum of square need not be the same as the total of sum of square of each individual
variables. This is either due to the interaction effect between the design variables or due to the
introduction of dummy variables, if any.
Percent contribution
The percent contribution of each design variable is the ratio of the sum of squares of a particular
design variable to the total sum of square of all the variables. This ratio indicates the influence of
the design variable over the performance parameter due to the change in the level settings.
Near optimal level value
In order to find the near optimal value of the objective function, a new experiment is conducted by
setting the near optimum level for each design variable. The near optimum level for any design
variable can be easily found from the mean values of all the level. The optimum level values can be
used as the initial value for further optimization problem.
ANOVA (Analysis of Variance) test
It may be noted from the previous sections that the significance of individual design variables can
be found from the percentage contribution. But it is not possible to categorically judge from the
contribution value whether 5% contribution is significant or not. Using analysis of variance
(ANOVA) approach, one can accept or reject a independent variable from the analysis given the
confidence level, . This can be done by conducting F-test [1].
As per the F-test, a variable is significant only if the ratio of mean sum of square of a variable (MSV)
to mean sum of square of error (MSE) is greater than the calculated F-value. The calculation of
MSV and MSE is based on the accumulation method [1] as given by the following equations.
18.
• the calculated F-value is based on the statistical
approach which obeys f-distribution with L-1
numerator degrees of freedom, N-L denominator
degrees of freedom and as confidence level. The
hypothesis for accepting or rejecting the
significance of a variable is given by the following
rules.
• Null Hypothesis (Ho) : The design variable is not
significant (5.11a)
• Alternate Hypothesis (Ha) : The design variable is
significant (5.11b)
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